January 13th, 2017, 2:01 pm

The model is:

[$]

dx(t) = [y(t) - \chi x(t)] dt + \sigma(t, x(t)) dW(t)\\

dy(t) = [\sigma(t, x(t))^2 - 2\chi y(t)] dt\\

x(0) = y(0) = 0\\

\sigma(t, x) = \alpha(t) + \beta(t)x\\

r(t) = f(0, t) + x(t)

[$]

Then, the pricing pde is

[$]

\partial_t v(t, x, y) + \frac{\sigma(t, x)^2}{2}\partial^2_x v(t, x, y) + (y - \chi x) \partial_xv(t, x, y) + (\sigma(t, x)^2 - 2\chi y)\partial_y v(t, x, y) - (f(0, t) + x)v(t, x, y) = 0

[$]

For the sake of simplicity you can try with [$]\chi = 0.01, \alpha = 0.007, \beta = 0.1[$] and [$] f(0, t) = 0.01[$].

Note that on this model, the zero coupon bond price is given by:

[$]

P(t, T) = \frac{P(0, T)}{P(0, t)} exp[-G(t, T)x(t) - \frac{G(t, T)^2}{2}y(t)]\\

G(t, T) = \frac{1 - e^{-\chi(T-t)}}{\chi}

[$]

This forumua is needed to compute the EURIBOR prices: [$]\delta L(t, \delta) = \frac{1}{P(t, t + \delta)}-1[$]

Maybe you can start to solve this PDE with the price of the product that pays [$]\delta L(t, \delta)[$] at [$]t + \delta = 20Y[$] with [$]\delta = 6M[$], ie the initial condition of the PDE is:

[$]

u(t, x, y) = \delta L(t, \delta, x, y) P(t, t + \delta, x, y)

[$]

Last edited by

VivienB on January 13th, 2017, 3:56 pm, edited 1 time in total.